Sum to Product Formulae in Trigonometry

The sum to product formulae in trigonometry provide identities that convert the sum or difference of sine and cosine functions into products of trigonometric functions of half-angles. These identities are essential in algebraic simplification and the solution of various trigonometric expressions.

Statement of the Sum to Product Formulae for Sine and Cosine

For all real $A$ and $B$, the following identities hold for the sum and difference of sine and cosine:

• Sum of sines as a product
• Difference of sines as a product
• Sum of cosines as a product
• Difference of cosines as a product

Result: The explicit formulae are given by:

$\begin{align*} \sin A + \sin B &= 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \\[10pt] \sin A - \sin B &= 2\sin\left(\frac{A-B}{2}\right)\cos\left(\frac{A+B}{2}\right) \\[10pt] \cos A + \cos B &= 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \\[10pt] \cos A - \cos B &= -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \end{align*}$

Each formula expresses a sum or difference of arguments as a product involving mean and half-difference or half-sum of $A$ and $B$.

Derivation Using Product to Sum Identities

To derive these, use the standard product to sum identities, valid for all real numbers:

• Expression of $\sin A \cos B$ as sums of sines
• Expression of $\cos A \sin B$ as difference of sines
• Expression of $\cos A \cos B$ as sum of cosines
• Expression of $\sin A \sin B$ as difference of cosines

Introduce the substitutions $p = A$, $q = B$ and set $a = \dfrac{p+q}{2},\ b = \dfrac{p-q}{2}$. Then, $a + b = p$ and $a - b = q$. Substitute these into product to sum identities and solve for sums or differences. For example,

$\sin p + \sin q = 2\sin\left(\frac{p+q}{2}\right)\cos\left(\frac{p-q}{2}\right)$,

by expressing $\sin p$ and $\sin q$ in terms of sum and difference, then algebraically combining and simplifying. The remaining formulae follow analogously using cosine sum and difference identities.

For a complete treatment of the proof structures, refer to the Product To Sum Formulae page.

Algebraic Interpretation and Variable Substitution

Let $A$ and $B$ be any real angles. The arguments in the product functions are always the mean $(A+B)/2$ and half-difference $(A-B)/2$, emphasizing the symmetry of the identities and their application to both positive and negative values of $A$ and $B$.

• Symmetry in arguments for $\sin$ and $\cos$
• Transformation under $A \leftrightarrow B$
• Negative angles and periodic behaviour

Simplification of Trigonometric Expressions Using Sum to Product Formulae

When an expression involves sums or differences of sines or cosines, these identities permit reduction to a product, facilitating further algebraic manipulation or integration.

For detailed study of other angle transformations, see Trigonometric Ratios of Compound Angles.

Illustrative Problems Based on the Sum to Product Formulae

Example 1: Express $\cos 4x - \cos x$ as a product.

Substitute $A = 4x,\ B = x$ into the identity for the difference of cosines:

$\cos 4x - \cos x = -2\sin\left(\frac{4x + x}{2}\right)\sin\left(\frac{4x - x}{2}\right)$

Simplify the arguments:

$\frac{4x + x}{2} = \frac{5x}{2},\quad \frac{4x - x}{2} = \frac{3x}{2}$

The result:

$\cos 4x - \cos x = -2\sin\left(\frac{5x}{2}\right)\sin\left(\frac{3x}{2}\right)$

Example 2: Evaluate $\sin 15^\circ + \sin 75^\circ$.

Use the sum of sines formula, $A = 15^\circ,\ B = 75^\circ$:

$\sin 15^\circ + \sin 75^\circ = 2\sin\left(\frac{15^\circ + 75^\circ}{2}\right)\cos\left(\frac{15^\circ - 75^\circ}{2}\right)$

$\frac{15^\circ + 75^\circ}{2} = 45^\circ$, $\frac{15^\circ - 75^\circ}{2} = -30^\circ$

$\cos(-30^\circ) = \cos(30^\circ) = \dfrac{\sqrt{3}}{2}$, $\sin(45^\circ) = \dfrac{1}{\sqrt{2}}$

$= 2 \times \dfrac{1}{\sqrt{2}} \times \dfrac{\sqrt{3}}{2} = \dfrac{\sqrt{3}}{\sqrt{2}} = \sqrt{\dfrac{3}{2}}$

Example 3: Simplify $\cos 8x + \cos 2x$ as a product.

Use the sum of cosines formula with $A = 8x,\ B = 2x$:

$\cos 8x + \cos 2x = 2\cos\left(\frac{8x + 2x}{2}\right)\cos\left(\frac{8x - 2x}{2}\right) = 2\cos(5x)\cos(3x)$

Example 4: Prove that $\dfrac{\cos 4x - \cos 2x}{\sin 4x + \sin 2x} = -\tan x$.

Apply the relevant formulae:

$\cos 4x - \cos 2x = -2\sin(3x)\sin(x)$
$\sin 4x + \sin 2x = 2\sin(3x)\cos(x)$

$\dfrac{\cos 4x - \cos 2x}{\sin 4x + \sin 2x} = \dfrac{-2\sin 3x \sin x}{2\sin 3x \cos x} = -\dfrac{\sin x}{\cos x} = -\tan x$

Example 5: Calculate $\sin 225^\circ - \sin 135^\circ$.

Using the difference of sines:

$\sin 225^\circ - \sin 135^\circ = 2\sin\left(\frac{225^\circ - 135^\circ}{2}\right)\cos\left(\frac{225^\circ + 135^\circ}{2}\right)$

$= 2\sin(45^\circ)\cos(180^\circ) = 2 \times \dfrac{1}{\sqrt{2}} \times (-1) = -\sqrt{2}$

Common Errors and Exam Identification Cues

A frequent mistake involves confusing the sum to product formulae with angle sum identities, such as $\sin(A+B)$. The sum to product identities operate on the sums or differences of two like functions (sine-sine or cosine-cosine), not mixed products or single expanded arguments.

• Confusing $\sin(A+B)$ with $\sin A + \sin B$
• Omitting the negative sign in $\cos A - \cos B$ formula
• Incorrectly evaluating negative angle trigonometric functions
• Not simplifying arguments consistently

For further comparative formulas, consult the Sum To Product Formulae and related entries on Trigonometry in compound angles.